Impact Factor: 0.763

Published since March 1, 2001

Advances in Geometry

ISSN: 1615-7168

Managing Editors: Theo Grundhöfer, Michael Joswig

Editorial Board: John Bamberg, Eiichi Bannai, Renzo Cavalieri, Izzet Coskun, Frank Duzaar, Patrick Eberlein, Graziano Gentili, Martin Henk, William M. Kantor, Gabor Korchmaros, Alexander Kreuzer, Jeffrey C. Lagarias, Thomas Leistner, Rainer Löwen, Kaoru Ono, Daniel Plaumann, Francisco Santos, Hendrik Van Maldeghem, Steven H. Weintraub, Richard Weiss

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Objective
Advances in Geometry is a mathematical journal for the publication of original research articles of excellent quality in the area of geometry. Geometry is a field of long standing-tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity; geometric ideas and geometric language permeate all of mathematics.

Topics

Finite geometry
Incidence geometry
Tits buildings
Combinatorial aspects of geometry and geometric combinatorics
Geometric group theory
Geometric aspects of Lie groups and of algebraic groups
Algebraic geometry
Topological geometry
Differential geometry
Convex sets and polytopes
Discrete geometry

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The journal was founded in 2001.

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Volume 23 Issue 1

January 2023

Publicly Available January 10, 2023

Frontmatter

Page range: i-iii

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January 15, 2023

Hilton–Milner results in projective and affine spaces

Jozefien D’haeseleer

Page range: 1-24

Abstract

In this article, we analyse maximal sets of k -spaces, in PG( n , q ) and AG( n , q ) with n ≥ 2 k + t + 3, that pairwise meet in at least a t -space. It is known that for both PG( n , q ) and AG( n , q ), the largest example is a t -pencil, i.e. the set of all k -spaces containing a fixed t -space. In this paper, we analyse the structure of the second largest maximal example in both PG( n , q ) and AG( n , q ).

January 17, 2023

Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two

Dae-Won Lee

Page range: 25-49

Abstract

In this paper, we give a complete classification of full exceptional collections, up to cyclic permutations, normalizations and mutations, on smooth toric Fano threefolds and fourfolds with Picard rank two. For such varieties, we find all the exceptional collections of maximal length and show that they are in fact full. This gives a partial answer to a conjecture in [29] and [32]. Moreover, such full exceptional collections essentially arise from Orlov’s theorem on projective bundles.

January 17, 2023

Ulrich trichotomy on del Pezzo surfaces

Emre Coskun, Ozhan Genc

Page range: 51-68

Abstract

We use a correspondence between Ulrich bundles on a projective variety and quiver representations to prove that certain del Pezzo surfaces satisfy the Ulrich trichotomy, for any given polarization.

April 18, 2022

Geometric characterisation of subvarieties of 𝓔₆(𝕂) related to the ternions and sextonions

Anneleen De Schepper

Page range: 69-106

Abstract

The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety 𝓔 6 (𝕂) over an arbitrary field 𝕂. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions 𝕆’ over 𝕂 (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties are linked to the Freudenthal–Tits magic square, and discuss how they would even fit in, when also allowing the sextonions and other “degenerate composition algebras” as the algebras used to construct the square.

January 17, 2023

Boomerang uniformity of power permutations and algebraic curves over 𝔽₂ⁿ

Sihem Mesnager, Ferruh Özbudak

Page range: 107-134

Abstract

We obtain the Boomerang Connectivity Table of power permutations F(x)=x2m−1 of F2n $F(x)={{x}^{{{2}^{m}}-1}}\text{ }\!\!~\!\!\text{ of }\!\!~\!\!\text{ }{{\mathbb{F}}_{{{2}^{n}}}}$with m ∈ { 3,n−12,n+12,n−2 }. $\left\{ 3,\frac{n-1}{2},\frac{n+1}{2},n-2 \right\}.$In particular, we obtain the Boomerang uniformity and the Boomerang uniformity set of F(x) at b∈F2n∖F2. $F(x)\text{ }\!\!~\!\!\text{ at }\!\!~\!\!\text{ }b\in {{\mathbb{F}}_{{{2}^{n}}}}\setminus {{\mathbb{F}}_{2}}.$Moreover we determine the complete Boomerang distribution spectrum of F ( x ) using the number of rational points of certain concrete algebraic curves over F2n. ${{\mathbb{F}}_{{{2}^{n}}}}.$We also determine the distribution spectra of Boomerang uniformities explicitly.

January 15, 2023

PL Morse theory in low dimensions

Romain Grunert, Wolfgang Kühnel, Günter Rote

Page range: 135-150

Abstract

We discuss a PL analog of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing through its level; it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong regularity are presented. In particular, we show that in dimensions d ≤ 4 a homologically regular point on a PL d -manifold is always strongly regular. Examples show that this fails in higher dimensions d ≥ 5. One of our constructions involves an embedding of the dunce hat into 4-space and Mazur’s contractible 4-manifold. Finally, decidability questions in this context are discussed.

January 17, 2023

Erratum for "Tropical superelliptic curves"

Madeline Brandt, Paul Alexander Helminck

Page range: 151-152

Abstract

We correct two errors in our paper Tropical superelliptic curves published in Advances in Geometry 20 (2020), 527–551. These corrections do not change the main results of the paper.

Ahead of Print / Just Accepted

Ahead of Print / Just Accepted

Volume 23 (2023)

Issue 1

Volume 22 (2022)

Volume 21 (2021)

Volume 20 (2020)

Volume 19 (2019)

Volume 18 (2018)

Volume 17 (2017)

Volume 16 (2016)

Volume 15 (2015)

Volume 14 (2014)

Volume 13 (2013)

Volume 12 (2012)

Volume 11 (2011)

Volume 10 (2010)

Volume 9 (2009)

Volume 8 (2008)

Volume 7 (2007)

Volume 6 (2006)

Volume 5 (2005)

Volume 4 (2004)

Volume 3 (2003)

Volume 2 (2002)

Volume 1 (2001)

5-year Impact Factor	0.708
Cite Score	1.1
Impact Factor	0.763
Journal Citation Indicator	0.67
Mathematical Citation Quotient	0.46
SCImago Journal Rank	0.513
Source Normalized Impact per Paper	1.011

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Editorial

Managing Editors

Theo Grundhöfer
Institut für Mathematik
Universität Würzburg
Emil-Fischer-Str. 30
97074 Würzburg, Germany
E-mail: advgeom@mathematik.uni-wuerzburg.de

Michael Joswig
Institut für Mathematik
Technische Universität Berlin
Sekretariat MA 6-2
Straße des 17. Juni 136
10623 Berlin, Germany

Editors

John Bamberg
Centre for the Mathematics of Symmetry and Computation
The University of Western Australia (M019)
35 Stirling Highway
Crawley, WA 6009, Australia

Eiichi Bannai
Faculty of Mathematics
Kyushu University
744, Motooka, Nishi-ku,
Fukuoka, Japan 819-0395

Renzo Cavalieri
Department of Mathematics
Colorado State University
Fort Collins, CO 80523-1874, USA

Izzet Coskun
Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
Chicago, IL 60607, USA

Patrick Eberlein
Department of Mathematics
University of North Carolina
CB 3250 Phillips Hall
Chapel Hill, NC 27599-3250, USA

Graziano Gentili
Dipartimento di Matematica
Universitá degli Studi di Firenze
Viale Morgagni, 67/a
50134 Firenze, Italy

Martin Henk
Institut für Mathematik
Technische Universität Berlin
Sekretariat MA 4-1
Straße des 17. Juni 136
10623 Berlin, Germany

William M. Kantor
Department of Mathematics
University of Oregon
Eugene, OR 97403-1222, USA

Gábor Korchmáros
Dipartimento di Matematica
Università degli Studi della Basilicata
Via Nazario Sauro 85
85100 Potenza, Italy

Alexander Kreuzer
Fachbereich Mathematik
Universität Hamburg
Bundesstr. 55
20146 Hamburg, Germany

Jeffrey C. Lagarias
Department of Mathematics
University of Michigan
530 Church Street
Ann Arbor, Michigan 48109-1043, USA

Thomas Leistner
School of Mathematical Sciences
The University of Adelaide
Adelaide SA 5005, Australia

Rainer Löwen
Institut fur Analysis/Topologie
TU Braunschweig
Pockelsstr. 14
38106 Braunschweig, Germany

Kaoru Ono
Research Institute for Mathematical Sciences
Kyoto University
Kyoto, 606-8502 Japan

Daniel Plaumann
Fakultät für Mathematik
Technische Universität Dortmund
Vogelpothsweg 87
44227 Dortmund, Germany

Francisco Santos
Departamento de Matemáticas, Estadística y Computación
Universidad de Cantabria
Av. de los Castros 48
39005 Santander, Spain

Steven H. Weintraub
Department of Mathematics
Lehigh University
Bethlehem, PA 18015-3174, USA

Richard Weiss
Department of Mathematics
Tufts University
503 Boston Avenue
Medford, MA 02155, USA

Technical Editor

Nils Rosehr
Institut für Mathematik
Universität Würzburg
Emil-Fischer-Str. 30
97074 Würzburg, Germany

(Deutsch)

Online ISSN: 1615-7168
Print ISSN: 1615-715X
Type: Journal
Language: English
Publisher: De Gruyter
First published: March 1, 2001
Publication Frequency: 4 Issues per Year
Audience: Researchers in the field of geometry

Advances in Geometry

Frontmatter

Hilton–Milner results in projective and affine spaces

Abstract

Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two

Abstract

Ulrich trichotomy on del Pezzo surfaces

Abstract

Geometric characterisation of subvarieties of 𝓔6(𝕂) related to the ternions and sextonions

Abstract

Boomerang uniformity of power permutations and algebraic curves over 𝔽2n

Abstract

PL Morse theory in low dimensions

Abstract

Erratum for "Tropical superelliptic curves"

Abstract

Geometric characterisation of subvarieties of 𝓔₆(𝕂) related to the ternions and sextonions

Boomerang uniformity of power permutations and algebraic curves over 𝔽₂ⁿ